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Grid Based Cancer Growth SimulationsDavide Alemani1, Francesco Pappalardo3, Emilio Mastriani2, Marzio Pennisi3 and

Santo Motta3

1EPFL, Lausanne, Switzerland, davide.alemani@epfl.ch

2Consorzio COMETA, Catania, Italy, mastriani@dmi.unict.it

3University of Catania, Catania, Italy, {francesco,mpennisi,motta}@dmi.unict.it

Abstract—In this work we introduce a newnumerical method to simulate tumor growth andimmune system, by applying an hybrid CellularAutomata-Lattice Boltzmann (CA-LB) approach.The CA-LB approach consists in using a CellularAutomata method to keep track of the immunesystem and the tumor shape and a Lattice Boltz-mann diffusion formulation to follow the variationof nutrient concentrations in the microenviron-ment of the tumor. The main aim is to use theCA-LB model to describe the interactions betweena growing tumor next to a nutrient source and theimmune system of the host organism. The startingpoints are two models independently developed.The first model is a reaction-diffusion model fortumor growth in presence of nutrients. It wasimproved by with the addition of a probabilisticCA model able to keep track of the immunesystem response to tumor growths and cell to celladhesion. The second model is based on a detaileddescription of the immune system at the cellularlevel with an agent-based method. The model issuccessfully used for cancer immunopreventionvaccine applications in mice. However, tumorgrowth is not modeled in detail and diffusivenutrient effects are missing. The COMETA GRIDinfrastructure was used as a key access pointto simulate several cancer growth scenarios thatreproduced different cancer growths observed inclinical cases.

Index Terms—Cancer, Agent based models, Hy-brid models, GRID

I. INTRODUCTION

IN this work we introduce a new numericalmethod to simulate tumor growth and

immune system, by applying an hybridCellular Automata Lattice Boltzmann (CA-LB)approach. The CA-LB approach consists inusing a Cellular Automata method to keep

track of the immune system and the tumorshape and a Lattice Boltzmann diffusionformulation to follow the variation of nutrientconcentrations in the micro-environment ofthe tumor. The main aim is to use the CA-LBmodel to describe the interactions between agrowing tumor next to a nutrient source andthe immune system of the host organism.The starting points are two modelsindependently developed in [1] and [2].The first model, [1], is a reaction-diffusionmodel for tumor growth in presence ofnutrients. It was improved by [3] and [4] withthe addition of a probabilistic CA model ableto keep track of the immune system responseto tumor growths and cell to cell adhesion.However, the immune system is described onlywith natural killer (NK) cells and cytolytic Tlymphocytes (CTL).The second model, [2], is based on a detaileddescription of the immune system at thecellular level with an agent-based method.The model is successfully used for cancerimmunoprevention vaccine applications in mice[5]. However, tumor growth is not modeled indetail and diffusive nutrient effects are missing.Both models have shown good agreement inreproducing different morphologies of growingtumors [4] and experimental data in naiveand vaccinated mice [2]. The main idea is tocombine the capabilities of both these modelsin order to qualitatively and quantitativelydescribe the micro-environment of a tumorgrowth, the diffusion of nutrients and theresponse of the immune system togetherwith possible immunoprevention vaccine

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applications.

THE MODELS

The reaction-diffusion model for nutrients

The nutrient evolution is based on a macro-scopic reaction-diffusion model [1], [3]. Ac-cording to [6], [7], nutrients are often classifiedinto two categories: essential and not essentialfor cell proliferation. An example of essentialnutrient is iron, essential for DNA synthesisand therefore for cell division. Other nutrientsaffect mainly the motility and death of cancercells [1]. We will call the former class ofnutrients as proliferative nutrients and denotetheir local concentration with N(x, t), while thelatter as survival nutrients and denote their localconcentration with M(x, t).The proliferative and the survival nutrients sat-isfy the following reaction-diffusion equationsrespectively:

∂N(x, t)

∂t= DN∇2N(x, t)− (1)

kN�T + I + λNC

�N(x, t)

∂M(x, t)

∂t= DM∇2M(x, t)− (2)

kM�T + I + λMC

�M(x, t)

The reaction-diffusion equations (1) and (2)describe the interaction between the nutrientsand the cells and the diffusion of the nutrients.The interaction nutrients-cells is described withlinear first order reactions.

The probabilistic CA model for tumor growth

The law of the tumor growth are based onprobabilistic rules already investigated and suc-cessfully used by [3], [1], [8]. In the modelwe distinguish between cancer (C) and necrotic(D) cells. The cancer cells are able to duplicate,move and die in the tissue, while the necroticcells are created when a cancer cells dies anddo not perform any action i.e. they remain ina necrotic state. They are not consumed by the

nutrients. Three parameters regulate the tumorgrowth: i) θdiv tunes the division action, ii) θmov

tunes the propagation action and iii) θdel tunesthe death action. Tuning accurately these pa-rameters allows us to reproduce different typesof tumor patterns.The probability laws are the following:

Pdiv = 1− e−�

NCθdiv

�2

(3)

Pmov = 1− e−C�

Mθmov

�2

(4)

Pdel = e−�

MCθdel

�2

(5)

The rules should taking into account the behav-ior of a real tumor. They mimic the fact that:

• Cell migration increases near the border ofthe tumor

• In region of low proliferative nutrient con-centration and with a high population ofcancer cells, cell division is inhibited andthe probability of cell’s death increases.

• In region of high survival nutrient con-centration and with a high population ofcancer cells, cell migration is enhancedand the probability of cell’s propagationincreases.

The LGCA model for the immune system

The LGCA method for the immune sys-tem is based on the SimTriplex simulator [2].SimTriplex is based on a modification of theCelada-Seiden framework [9].SimTriplex mimics at the cellular level thebehavior of immune cells of vaccinated as wellas naive mice. This is a generalized cellularautomaton and it is specialized in modelingmammary carcinoma, Triplex vaccine and im-mune system competition.In order to simulate immune system competi-tion, the following entities and interactions wereadded in SimTriplex:

1) the cancer cells (C) which encode theirtumor associated antigens (TAA); theyinteract with Ab, TC and NK.

2) the vaccine cells (VC) which includeTAAs, IL-12 and allogenic MHCI; VC

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interact like C, with Ab, TC and NK, butthe affinity function is modified by thepresence of the two adjuvants.

3) the natural killer (NK) cells which do notencode receptors; they interact with Ab,VC and C.

A detailed explanation of the SimTriplex modeland the symbols used in the list above is givenin [2].The main difference between the original, see[2], and the modified versions of Simtriplex isin the regulation of tumor growth. The threeprobabilistic functions, given by equations (3),(4) and (5), were added in the original ver-sion of SimTriplex, giving raise to a modifiedSimTriplex version. Most of the models arenaive because they offer a simplistic descriptionof biological systems. Now we have a capacityto link genetic, functional and physical aspectsof these models. The key is to determine amore complete sets of probabilistic rules thatrepresent various actual situations and multipleforms of tumors and hallmarks (thing that maketumors what they are).

REALISTIC SIMULATIONS OF TUMORGROWTH

After validation, we have performed severalsimulations with the aim to reproduce realistictumor patterns under different conditions. Wehave performed qualitative comparison with ac-tual melanoma using the Linux cluster at theCancer Vaccine Center of the Harvard MedicalSchool in Boston (USA). We ran the numericalalgorithm with the aim to reproduce typicaltumor patterns from actual melanomas. Thevalues of the parameters were chosen to be asmore realistic as possible based on values givenin the literature. We have done four differentsimulations that aim to represent four types oftumors.

The results of each simulation are reproducedin figures 1, 2, 3 and 4.Figure 1 shows a typical configuration of tumorwith uneven edges. We notice that the immunecells align themselves at the border of the tumormass, with a tendency to accumulate close to the

nutrient. Necrotic cells tend to accumulate nearthe source of nutrient, as well. Most likely, thismeans that these are sites where cancer cellsappear and then instantly die. Indeed, necroticcells do not consume nutrients, therefore thereare no reason that they get close to the source ofnutrient (in our case the blood vessel) unless wemake the hypothesis that cancer cells migrateclose to the source of nutrient attracted by itspresence and then die, becoming necrotic cells.The tumor shows a finger-like pattern. It isevident that the presence of the nutrient sourceis a stress factor for the tumor mass. Tumorcells are attracted by the presence of food, thenthey die and fractal-like patterns appear. It isinteresting to observe that the cancer cells didnot exhibit a finger-like pattern but they preferto accumulate at the border of the tumor massand close to the source of nutrient, generating acancer proliferative layer. Indeed, the presenceof such a proliferative layer is known in theliterature [10], [11], [12].Figure 2 shows the results of a simulation wherethe migration probability is much larger thanthe simulation shown in figure 1. The finaldistribution shows a tumor with less borderirregularity but more compact surface that thatshown in figure 1. We notice that necrotic cellsare absent. We argue that this kind of tumorare benign, because the necrosis process didnot start. Cancer and immune cells show thesame tendency as depicted in figure 1. Ac-cumulation of cancer cells near the source ofnutrient determines an asymmetric distributionless pronounced than that shown in figure 1,probably due to the absence of necrotic cells.This can be explained considering the differentnature of cancer and necrotic cells. Necroticcells do not perform any action, they are likeinert cells, while cancer cells are living cells thatare “hungry” of food much more than normaltissue cells. As a consequence, all cancer cellsare attracted to the nutrient source. If duringthis attraction process, some of the cancer cellsdie and become necrotic cells, a discontinuityappears: where necrotic cells appear, the cancercells present in their neighbor find themselves

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surrounded with more food, thus they slowdown their attraction process to the nutrientsource creating irregularity in the tumor shape.Comparison of figure 2 with figure 1 shows thatthe presence of the nutrient plays an importantrole in determining the shape of a tumor andconsequently its benign or malignant nature.

In the simulations shown in figures 3 and

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Fig. 1. Simulation A. Parameters are set as follows: Kn =10−5, θdiv = 10−3, θmov = 105, θdel = 10−2

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Fig. 2. Simulation B. Parameters are set as follows: Kn =10−5, θdiv = 10−3, θmov = 103, θdel = 10−1

4 the tumor growth model parameters are thesame, but cancer cells are more aggressive insimulation C (the rate constant kN is largerin simulation C than in simulation D). As aresult, the tumor shape are similar and they

show necrotic cores and border irregularity. Thepeak number of necrotic cells is about 104 insimulation C and about 5 · 104 in simulation D.Among the total number of cancer and necroticcells, in both simulations 37% are cancer cellsand 63% are necrotic. However, although bothsimulations stop after 50 time steps, the totalnumber of tumor cells (cancer and necrotic) are106 in simulation C and 4 · 106 in simulationD. The less amount of tumor cells in simulationC can be explained by considering the moreaggressivity of cancer cells in consuming nu-trients. Cancer cells in simulation C consumenutrient more rapidly than in simulation D,therefore, after a same amount a time, in someregions there will be less nutrient availablefor consumption in simulation C than in sim-ulation D. This fact determines a decrease ofthe division probability of cancer cells Pdiv insimulation C and this may explain why the totalnumber of tumor cells is lower in simulation C(figure 3) than that in simulation D (figure 4).In figure 5 we show all the four simulations A,

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B, C and D and a possible link with four differ-ent types of real tumor found in the literature.

CONCLUSIONS

The main conclusion of our work reproducedin this paper, can be summarized as follows:

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Fig. 4. Simulation D. Parameters are set as follows: Kn =10−7, θdiv = 10−3, θmov = 105, θdel = 10−2

Fig. 5. The results from the four simulationsA, B, C and D are related with four differentreal tumor shapes taken from internet at the follow-ing website: http://www.southcoastmedcenter.com/graphics-/lab/abcd melanoma.jpg. A red-yellow color scale has beenused to better match the real tumor shapes.

• Our model is able to reproduce early stageavascular tumors. In particular, it is ableto correctly predict the formation of threemain stages: i) the necrotic cores madewith necrotic cells, ii) the quiescent layermade essentially with cancer and necroticcells and iii) the proliferating layer madewith cancer cells.

• Our model is able to qualitatively re-produce finger-like tumor patterns. Bothnecrotic and cancer cells show this type ofshape. By tuning the parameter that defines

the probability of necrosis, it is possibleto control the number of necrotic cells. Inparticular, when the necrosis probability isreduced, we do not observe the formationof necrotic cells, but only cancer cells thataccumulate near the source of nutrient.

• Our model is able to qualitatively repro-duce tumor shapes with border irregular-ity, by decreasing the consumption rateof nutrient by the cells. It is interestingto see that more than one necrotic coreappear, that is an indication of malignancyin melanomas.

ACKNOWLEDGMENT

This work was supported under the EC con-tract FP6-2004-IST- 4, No.028069 (Immuno-Grid). This work makes use of results producedby the PI2S2 Project managed by the ConsorzioCOMETA, a project co-funded by the ItalianMinistry of University and Research (MIUR)within the Piano Operativo Nazionale ”RicercaScientifica, Sviluppo Tecnologico, Alta For-mazione” (PON 2000-2006). More informa-tion is available at: http://www.pi2s2.it andhttp://www.consorzio-cometa.it.

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[7] M. Scalerandi, G.P. Pescarmona, P.P. Delsanto, andB. Capogrosso Sansone. Local interaction simulationapproach for the response of the vascular system tometabolic changes of cell behavior. Physical Review

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Davide Alemani graduated fromthe University of Milan, Italy in2001 with a Masters degree inphysics. His thesis was based onthe theory and applications ofclassical and modern mathemati-cal methods to solve partial dif-ferential equations in applied sci-ences. He continued his career atthe Department of Chemistry ofthe University of Geneva obtain-

ing a PhD in computational science for environmentalchemistry in 2007. He worked on the application of theLattice Boltzmann Method to compute metal fluxes inenvironmental systems. During this time he successfullydeveloped a numerical code (MHEDYN) which is beingadopted by the Chemistry community. In 2008, he wasgranted a scholarship by the Swiss National Foundation inwhich he spent 12 months at the University of Queensland,Australia, working on a research project (IMMUNOGRID)in computational modeling for the tumor growth and im-mune systems. He is currently pursuing a post-doctorate inthe field of molecular dynamics at the EPFL in Lausanne,Switzerland.

Francesco Pappalardo earned hismaster and PhD. degree from Uni-versity of Catania, in 2000 and2004 respectively. Actually he isholding a postdoctoral fellow andhe is temporarily assigned to theprofessorship of Mathematics atthe University of Catania. From2001 to 2004 he did researchin computer operating system se-curity. In 2008 he was visiting

scholar at Dana-Farber Cancer Institute in Boston (USA)while in 2006 he visited the Molecular ImmunogeneticsLabs, IMGT in Montpellier (France). His major researchareas are on modeling human and mouse immune systemresponses involved in several pathologies, including viralinfections, tumors, and atherosclerosis. This effort leadedand actually leads to both a better understanding of theimmune system, and to the application of these modelingapproaches to vaccine research. Up to now, Francesco Pap-palardo published more than 40 reviewed research papers.He serves the scientific community as editorial board mem-ber, reviewer and program committee member for majorjournals and conferences in the area of computer scienceand computational biology.

Marzio Pennisi graduated inComputer Science in March,2006, at University of Catania,Italy. In December of the sameyear, he won the competitionfor Ph.D. course of study atthe University of Catania,Italy. He is actually studyingmodels, mathematical methodsand computer simulation of theimmune system. In particulary he

is working on models of mammary tumor growth, diffusionof metastasis and therapy. His present scientific interestsrange from BioMaths BioComputing and BioInformaticsImmunomics, evolutive algorithms, constrained andunconstrained optimization.

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Santo Motta was born in Romein 1946. He obtained “Laurea”in Physics at the University ofCatania (1970) and its M.Sc. inApplied Mathematics at the Uni-versity of London, Queen MaryCollege (1971). Actually he is as-sociate professor of MathematicalPhysics at the Faculty of Phar-macy of the University of Cataniaand he is member of the Dept. of

Maths & Computer Science of the University of Catania.Presently he is interested in computational models of thecancer immune system competition induced by an in vivotested immune prevention vaccine. Its present scientificinterests are BioMaths, BioComputing and BioInformatics,Immunomics. Previously he was interested in numericalmethods for transport equation, Non linear waves in classicand relativistic fluid?dynamics, General Relativity and Cos-mology, Solar Physics. Up to now, Santo Motta publishedmore than 80 reviewed research papers and he is memberof editorial board and reviewer of several internationaljournals.